A Six-Periodic Zero of Focal Outer-Antipedal Area

For a periodic elliptic billiard, form the polygon of intersections of consecutive boundary tangents, then its antipedal with respect to either focus. Classical central symmetry implies equality of the two signed antipedal areas for effective even periods. Their quotient is therefore one where defined, but it need not be defined everywhere. We give an explicit convex billiard of least period six on an ellipse of aspect ratio 1+sqrt(3) for which both areas vanish, while all antipedal vertices are finite, all edges are nonzero and the vertices are not collinear. A direct coordinate calculation gives the signed area of an axial six-periodic family and isolates the zero exactly. This is a denominator obstruction, not a counterexample to equality on the quotient's natural domain. The earlier original-polygon antipedal zero at aspect ratio 2 is distinguished and credited. Source record: AMR-050-0036 (raw ID 5100036), invariant k608 in the frozen Hugging Face dataset ulamai/UnsolvedMath v1.6.0. Supporting lines and ordered signed area are used. The effective-even scope includes admitted coprime stars and repetitions of even-primitive orbits; no all-phase zero, odd-primitive repeated-list extension, hyperbolic-caustic extension or unqualified whole-source resolution is claimed. A portable Python standard-library exact certificate accompanies the analytic proof. This is a self-audited, AI-assisted, unrefereed preprint. Classical symmetry and the prior original-polygon zero are credited; novelty remains undetermined after a bounded search. No independent human review, proof-assistant verification or absolute-priority certification is asserted.